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 /*  batch file for maxima 
 uses :
 - symmetry around horizontal ( 0X ) axis
 - Psi_M function to map conjugate plane to parameter plane
 - jungreis algorithm to 
 time :
 263 sec for jMax:100; and iMax:800;
 4000 sec for jMax:400
 
 */ 
 
 
kill(all);
remvalue(all);


/* ============== functions ============================================================================*/

/* ------------ computes b coefficient of Jungreis function ----*/
 betaF[n,m]:=block
 (
 [nnn:2^(n+1)-1],
 if m=0
 then 1.0
 else if ((n>0) and (m < nnn)) 
  then 0.0
  else (betaF[n+1,m]- sum(betaF[n,k]*betaF[n,m-k],k,nnn,m-nnn)-betaF[0,m-nnn])/2.0
 )$
 b[m]:=betaF[0,m+1]$
 /* -------------------------------*/
 /* Power of w to j */
 wn[w,j]:= if j=0 then 1 else w*wn[w,j-1]$
 

/* ---------Jungreis function ;  c = Psi_M(w) ----------------------------- */
 Psi_M(w):=w + sum(b[j]/wn[w,j],j,0,jMax)$

 /* exponential for of complex number with angle in turns */
 GiveCirclePoint(t):=R*%e^(%i*t*2*%pi)$ /* gives point of unit circle for angle t in turns */
 

compile(all);

start:elapsed_run_time ();
 
 jMax:1000; /* precision  = proportional to details and time of computations  */
 iMax:2000; /* number of points to draw */

/* --- w-plane -----------------------------------------*/
 

/*-------------- unit circle ------------*/
 R:1;
 circle_angles:makelist(i/iMax,i,0,iMax/2)$
 CirclePoints:map(GiveCirclePoint,circle_angles)$
 
/* -------------------------parameter plane = c plane -----------------------------------*/
 MPoints:map(Psi_M,CirclePoints)$ /* Mandelbrot set points */
 /* add points below horizontal axis */
 for w in CirclePoints do CirclePoints:cons(conjugate(w),CirclePoints)$
 for c in MPoints do MPoints:cons(conjugate(c),MPoints)$
 
/* time */
 stop:elapsed_run_time ();
 time:fix(stop-start); 
 
/* ---------------- draw *--------------------------------------------------------------------------*/

 path : ""; /* pwd  ; if you put here working directory name then graphic file will be saved in that dir */
 FileName:concat(string(jMax),"jung", string(iMax))$ /* without extension which is the terminal name */


 load(draw); /* Mario Rodríguez Riotorto   http://www.telefonica.net/web2/biomates/maxima/gpdraw/index.html */
 draw(file_name = concat(path,FileName),
  dimensions=[2000, 1000],
  terminal  = 'svg,
  columns  = 2,
  gr2d(title = " unit circle {w:abs(w)=1} ",
  point_type = filled_circle,
  points_joined =true,
  point_size    =  0.34,
  color         = red,
  points(map(realpart, CirclePoints),map(imagpart, CirclePoints))
  ),
  gr2d(title      = " Image of circle under Psi_M(w) ",
  points_joined =true,
  point_type = filled_circle,
  point_size    =0.34,
  color         = blue,
  points(map(realpart, MPoints),map(imagpart, MPoints))
  ) 
 );

1. ↑ Drawing Mc by Jungreis Algorithm